As seen in Section 11.1.3, power factor reaches its optimal value only when the input current is proportioned to the input voltage. The conclusion was reached that if the
line voltage was a distorted waveform, then the line current should be distorted as well, containing harmonic currents proportional to the harmonics of the distorted voltage. But what is the effect on the utility line of these harmonic currents? Might it not be better from a system-wide point of view to force each device on the line to draw a sine-wave current in phase with the fundamental of the line voltage? Such a scheme would require an internal sine reference locked to the fundamental of the voltage, and would be much more complicated than a circuit generating proportional input current.
Fortunately, it is easy to show that the simpler solution—proportional input current—
is better for both the individual power converter and for the power line as a system. It is true that a distorted line voltage leads to harmonic currents in converters emulating resistive loads. However, these harmonic currents tend to cancel the very harmonic currents that cause the voltage distortion in the first place.
Consider the power system of Fig. ll.l(a). The utility line is represented by an ideal voltage source vg producing a sine-wave voltage, in series with an impedance Zg. The line, with voltage υ∣, supports two loads. The first load draws a distorted current i<ust, producing distortion in the line voltage. The second load is an ac-dc power converter, drawing a current ipc which is either a fixed sine wave or proportional to the line voltage.
In the latter case the power converter appears to the line as a resistor of value Rem.
At each harmonic frequency, the power source and its impedance can be replaced with a Norton equivalent circuit. The equivalent circuits for the fcth harmonic (fc > 1) are shown in Figs. 11.1(b) and (c). A tilde (~) denotes a phasor quantity and the k subscript emphasizes that these phasors are at k times the fundamental frequency.
At each harmonic frequency, the source impedance Zgι∣e is assumed non-zero. Since Vgιk is zero for every k > 1 (the voltage υg is undistorted), the current source labeled Va,⅛∕¾,⅛ is an open circuit. For the case where the power converter draws a pure sinusoid, shown in Fig. ll.l(b), Ipcι∣e = 0 for k > 1 and this current source is also an open circuit. The entire harmonic current of the distorting load, Iditt,k must pass through the impedance Zgjt generating a harmonic voltage — idist,k⅞Λ *n the line voltage.
If instead the power converter draws a current proportional to the line voltage, the equivalent circuit of Fig. ll.l(c) applies. Here the current source F⅛,*∕∙¾,jfc is still an open
Power Converter
(b)
½,fc — ïdist,k (∙⅞,fc∣∣∙¾m∙^
(c)
Figure 11.1: A power system including a source of distorted current and a power con
verter (a). Model of the system at the λth harmonic frequency when the power converter draws a sinusoidal current (b) and when the power con
verter emulates a resistor (c).
circuit, but the power converter now appears in the equivalent circuit as a resistor Rem∙
The harmonic current Idist,k now flows in the parallel combination of Rem and Zg. The magnitude of the impedance of this parallel combination is always less than that of Ziι∣c alone (since Rem and the real part of Zβι∣e are both positive). Hence the magnitude of the harmonic voltage produced by the distorted load is less when the power converter draws a proportional current.
Note that when the resistor-emulating power converter draws harmonic currents from the power line, it still achieves unity power factor, because the harmonic currents are directly proportional to harmonics of the distorted line voltage.
Usually the line impedance Ziι∣c is small at frequencies of interest, dominating the parallel combination of Zβt
∣
t and Rem∙ The effect of any single power converter in attenuating voltage distortion is therefore small. However, if many devices on the power line have proportional currents, their combined effect is that of a small Rem and the effect on the line may be significant. In effect, the distorted load draws its harmonic currents from the power converters instead of through the line impedance Zg. The result is a reduction of harmonic voltage relative to the case where the power converters draw only fundamental current.
The important point is that drawing a proportional current from the power line is beneficial not only to the load, which achieves unity power factor, but also to the power line. Every device that emulates a resistor makes some contribution to removing distortion from the line voltage.
Chapter 12
Passive Current-Shaping Methods
Some circuits for improving input-current waveforms of ac-dc converters use only passive components—reactances and semiconductor rectifiers. These passive circuits—
essentially filters—are valued for their simplicity and ruggedness. They avoid the rel
atively fragile semiconductor switches and complicated control circuits found in active current-shaping circuits.
Passive schemes have the drawback of excessive size and weight compared to equally effective active circuits. The passive circuits are incapable of providing regulation against load variations and are also sensitive to variations in the line frequency. Finally, most passive circuits provide only an approximation to a sine-wave input current, and the maximum achievable power factor is something less than unity.
In many applications, however, the attractions of simplicity and robustness outweigh these drawbacks. At very high power levels, for example, fast-switching with active devices is impractical. Thus when electric utilities transmit large amounts of power over dc links, passive filters are used to improve the waveforms on the ac side of dc-ac converters [24, Sec. 8.8]. Another application where passive filters have an advantage is in the rare instance of a high-frequency ac supply. In passive schemes, the reactances are sized in inverse proportion to the line frequency, while in active schemes the switching frequency must be some factor above the line frequency. For fixed performance, passive filters shrink as the line frequency increases, but active circuits have to switch much faster. At some point the passive filters will be favored. In the 20 kHz power distribution proposed for the space shuttle [25, pp. 300-303], for instance, any current-shaping for ac-dc power converters would likely be passive.
The inductor-input filter studied in Section 12.1 is probably the most common pas
sive filter used. The filter is simple and relatively cheap. Its main drawbacks are a 0.90
Figure 12.1: The inductor-input filter.
upper limit on power factor, and discontinuity of the input current for reasonably large power factor. The resonant-input filter of Section 12.2, which uses a single series-resonant circuit, is an alternative way to passively shape input current. Although heavy and ex
pensive, it shows that unity power factor is possible using just a few passive components and no active devices. More often used as a voltage regulator, the ferroresonant trans
former can also be considered as a passive current shaper. It too has a resonant circuit that encourages sinusoidal input current, as seen in Section 12.3. Finally, power factor can be improved by removing harmonic currents using tuned filters, a method considered in Section 12.4. A comparison shows that a set of tuned filters offers considerable size and weight savings over a resonant-input filter.
12.1 Inductor-Input Filter
A simple way to improve the input-current performance of a capacitor-input filter is to place an inductor in series with the output of the bridge rectifier, as shown in Fig. 12.1.
This configuration is called the inductor-input filter. The inductor promotes continuity of the input current, broadening the short pulses drawn by the capacitor-input filter and increasing the power factor.
Several analyses of the inductor-input filter appear in the literature [26],[27]. The analysis presented here emphasizes simultaneous exposure of the effects of inductance L, fine (radian) frequency ω∕ or line period T∕, and load R ≡ V∕I∙ The dimensionless
parameter Kι, defined by
cüi L 2L
Kl =
(12.1)πR RTl
serves this purpose well and supplants the normalized units of [26] and [27]. The subscript on Kι means that Kι involves the line frequency. The output voltage V of the inductor-input filter is assumed constant for purposes of analysis.
The filter operates in either the continuous conduction mode (CCM), in which the rectifier bridge always conducts, or in the discontinuous conduction mode (DCM) when the bridge is OFF during some portion of the line period. This is analogous to PWM dc-dc converters which have both continuous and discontinuous conduction modes. In fact, the parameter Kι of Eq.(12.1) is defined in exactly the same way as the parameter Ks ≡ 2L∕RTs found in dc-dc converter analysis. Where Kι is defined using the line period 7}, Kg uses the switching period Tg ■ Both parameters are measures of the position of the operating points of their respective circuits relative to the CCM/DCM boundary.
12.1.1 Continuous Conduction Mode
In the continuous conduction mode, the bridge rectifier is always conducting, and the voltage va is a rectified sine wave with peak value ½. Since the average voltage across the inductor must be zero, the output voltage is
V = -½, (12.2)
%
where ½ is the peak value of the line voltage. Integration of the voltage across the inductor gives the shape of the inductor current ig. The constant of integration is chosen to produce average current I, yielding
ig(θ) = ∣l + 2¾ -cos0 - — 1 . (12.3)
<jJιh 7Γ J
The rms value of the line current is the same as the rms value of ia,
iδ∙ms = ∖A + ’ (12'4)
found by integrating the square of Eq.(12.3). The power factor follows directly from Eqs.(l2.2) through (12.4), and is a function only of Kγ.
2∖f2 1 0.90
(12.5) PF =
’ √1 + (⅛ - ⅜) W a √ 1 + (°°7δ∕¾) 2 '
The displacement factor, cos≠ι, is found by calculating the Fourier sine and cosine coefficients of the input current, the odd extension of Eq.(l2.3) into the full line period, and is given by
1 1 (12.6)
COS<∕>ι =
√ι + (i-i),∕κi, √ι + ⅛)i
The boundary of CCM, referred to as critical conduction, occurs when the current of Eq.(12.3) just reaches zero at some point during a cycle. Equation (12.3) can be differentiated to find that the minimum current occurs when
θ = sin-1 ≈ 40° . (12.7)
The value of Kι that makes the current exactly zero at this point is found to be
-K),crit = sin-1 + cos sin-1 — 1∣ « 0.1053 . (12.8)
12.1.2 Discontinuous Conduction Mode
In the discontinuous conduction mode (DCM), the rectifier bridge is OFF during some fraction of each cycle. Analysis of this mode of operation is more difficult than for CCM.
The power factor and its displacement-factor component cannot be written explicitly in terms of the parameter Kι, but they are nevertheless functions only of this parameter.
Here, an implicit equation for the turn-ON and turn-OFF angles of the rectifier bridge is derived. The power factor, output voltage, and the parameter Kι are then expressed as explicit functions of these angles.
The analysis is divided into two cases, corresponding to the two inductor current waveforms of Fig. 12.2. In both cases, conduction of the bridge rectifiers begins when the input voltage rises to meet the output voltage. The angle 0χ is defined as the value of θ at this instant, giving the relation
— = sin θi . V
vl (12.9)
Case I
Case IIFigure 12.2: Inductor-current waveforms for the discontinuous mode of operation of the inductor-input filter.
The angle 02 is defined as the value of 9 at the instant the inductor current reaches zero. For case I of Fig. 12.2, Θ2 lies in the same half cycle of the line voltage as 9γ, that is, Θ2 < π. Case II occurs when the inductor current extends into the next half cycle, and 02 > *∙
The inductor current for case I is given over the interval 0i < 9 < Θ2 by ig(9) — —Γ [cos *1 ~ cθs 9 — (9 — 0x) sin 0ι]Vι
The cut-off angle 02 is found by solving
cos 0ι — cos 02 — (02 — 0ι) sin 0χ = 0
(12.10)
(12.11) This transcendental equation must be solved before Kι or the power factor can be found.
Once 02 is known, however, the average and rms inductor currents can be calculated, along with the output voltage and Kι'.
Kl =
*i>avg —
•2
*girr∏Λ
~^2"ip01 {sin “ 2^2 “ 01)2J ~ sin ^2 + - *l) c°s θl}
—- ιrKι sin 0χV, ω∣L
(j⅛) ; {s⅛ 0<9≈ - 9>> sin"' - ∞sβ∙ιs+coss 90 + ∣(02 - 0ι} + ∣ sin 202 + 2(02 — 0χ) sin 0j sin 02
— j sin 0ι cos 02 — 2 sin(02 — 0ι) ∣
(12.12) (12.13) (12.14)
V VI
COS θi — COS 02
(02 — 0l) The power factor is given by
and the displacement factor by
(12.15)
PF =
COS≠1 =
√2tg,av8sin 0ι
*J,ΓH1S
h
(12.16)
∖∕0ι + bl
where the Fourier coefficients of the fundamental, αj and δχ are given by οχ = - lsin(02 - 0χ) - (02 - 0ι)(j + sin01sin02)
7Γ t
— I (sin 202 — sin 20χ) j 2
(12.17)
(12.18)
bi = [(02 — 0χ) sin 0χ cos 02 — cos(02 — 0ι) (12.19)
+ 1 + ∣(cos2 02 — COS2 0ι)J
The boundary between cases I and II occurs when 02 = τr. The value of 0χ cor
responding to this from Eq.(l2.11) is approximately 0.8105 radians, about 46 degrees.
The conduction parameter Kι is approximately 0.051 at this boundary. The boundary between cases I and II also marks the change between continuous and discontinuous line current. For Kι > 0.051, the line current is discontinuous.
For case II, analysis proceeds just as in case I except that for 0 > jγ the voltage from the bridge is —Vj sin 0, to account for the rectification of the line voltage. The inductor current now follows Eq.(12.10) for 0χ < 0 < π, but for π < 0 < 02,
ti(0) — [(2 + cos 0ι + 0ι sin 0χ) + cos0 — 0 sin 0χj . (12.20) The cut-off angle 02 solves
cos 0ι + 0ι sin 0ι + cos 02 - 02 sin θ1 + 2 = 0 (12.21) The average inductor current is still given by Eq.(l2.13), but the conduction parameter, rms inductor current, and conversion ratio for case II are given by
κ~ = i⅛MH(i>-v]
+2(02 — jγ) + (02 — 01) cos 0ι + sin 02}
(12.22)
’«·""· = (⅛) ï {3⅛ (,("2 - ’I’ si"tl -∞s9l]s+ <12∙23) + ($2 — 0ι)( j - 2sin sin #2) + 2sin(02 — #i)
+ 4(sin 02 - sin 0ι) + ∣(sin 20χ — sin θi)
+4(02 — π)(l + cos 0i + 0ι sin 0ι — ⅛(02 + π)sin0ι)j
ÏÏ = (02^^^0ι) (2 + cosffl + cos02) ∙ (12.24) The power factor for case II is evaluated with Eqs.(12.16) and (12.13), using the expressions Eqs.(12.22) and (12.23) for Kι and t'i,rms> respectively. The displacement factor for case II is found from Eq.(12.17) with the following expressions for the Fourier coefficients:
a1 = — ∣-sin(02 - 0ι) - ∣(sin202 - sin20ι) + ∣(0i - 02) (12.25) + (^2 — *ι) sin 0ι sin 02 — 2 sin 02 + 2 sin 0χ |
⅛ι = — {—(02 — 0ι) sin 0ι cos 02 + 2 cos 0ι + 2 cos 02 2 (12.26) 7Γ
+ cos(02 — 0ι) + I (cos 202 — cos20ι) + 3 — 2(π — 0ι)sin0ι^ .
The above expressions are unwieldy, and a plot of the results in the following section will convey much more information. One point to note in the expressions for conversion ratio and power factor is that these quantities may be considered functions only of the conduction parameter Kι. From Eqs.(12.1l) and (12.21), the angle 02 is a function only of 0ι. Since Eqs.(12.12) and (12.22) are one-to-one, 0ι and 02 may be considered functions of Kι alone. The conversion ratio, the power factor, and the displacement factor, all expressed here as functions of 0χ and 02, are therefore functions of Kι alone.
12.1.3 Interpretation
Because power factor and its displacement factor component are functions of Kι alone in both CCM and DCM, a single plot is sufficient to show the effect of all operating conditions. In Fig. 12.3, the power factor and displacement factor are plotted versus Kι on a log scale. The conduction parameter Kι, it will be recalled, is the ratio ωιL∕πR.
1 .9 .8 . 7
.6 DCM I→
DCM II ≤- CCM
5 L_l---L_
.001 .01 10
Kj
. 1 1
Figure IS.8: Power factor, displacement factor cosφι, and conversion ratio M ≡ V∕¼ for the inductor-input filter.
The plot of Fig. 12.3 therefore includes the effects of line frequency, filter inductance, and effective load resistance.
The conversion ratio V∕½ is also included in the plot of Fig. 12.3. In CCM the conversion ratio is constant at the value 2∕π. As Kι is reduced and the filter enters DCM, the conversion ratio rises. In the limit Kι → 0, the output voltage equals the peak value of the input sine wave and the conversion ratio is unity.
Maximum power factor with the inductor-input filter is 0.90, achieved as Jf∣ → ∞.
As Kι is reduced, the power factor falls to a value of only 0.732 at the critical point. As Kι falls below K∕ιtrit1 the power factor initially increases, a somewhat surprising result.
If K[ declines still further, the power factor decreases monotonically.
The curious behavior of the power factor is explained by the waveforms of Fig. 12.4 and the displacement factor curve of Fig. 12.3. Figure 12.4 shows the waveshape of the line current (“unfolded” inductor current) as Kι varies. For large jK^j, as in Fig. 12.4(a), the line current is nearly a square wave. The fundamental component is in phase with the line voltage; distortion alone limits the power factor to 0.90. As Kι falls, some ripple appears across the previously flat tops of the the square wave, as shown in Fig. 12.4(b), and the fundamental lags slightly.
Figure 12.4: Line-current waveforms (solid curves) for the inductor-input filter as Kι changes. The dashed curves represent the line voltage.
Note the discontinuities in the line current in Fig. 12.4(a) through Fig. 12.4(c). “Con
tinuous conduction” of the inductor-input filter means only that the bridge rectifier is always conducting. The line current is actually discontinuous (in the sense that it has vertical jumps) all through CCM and into part of DCM. Only when the circuit enters case II of DCM does the line current become continuous.
Critical conduction is illustrated in Fig. 12.4(c). From Eq.(12.7), the input current touches zero when θ ≈ 40°. Since the current is not zero when the rectifier bridge switches, at θ = 0 and 0 = π, the line current is discontinuous. The distortion is small, but phase displacement of the fundamental keeps the power factor relatively low. As Kl falls to its critical value, shown in Fig. 12.4(d), the distortion of the current increases, but at the same time the displacement factor decreases. The combined effect is an increase in power factor.
Eventually, for very small Kι, the line current becomes a narrow, nearly centered spike, as in Fig. 12.4(e). The fundamental is almost in phase in this case, and the low
Figure 12.5: The resonant-input filter.
power factor is a result of distortion alone.
The inductor-input filter offers the advantages of passivity, simplicity, and ruggedness.
Undesirable aspects of the filter as an input-current shaping method are the large, heavy inductor, the lack of output-voltage regulation, the fact that power factor varies with the load, the discontinuity of the input current, and the 0.90 upper Emit on power factor.